The Wrong Way to Use a Signed Distance Function (SDF)

Summary

Disclaimer: there’s nothing wrong with using a sdf this way.Recently, my good friend Mike Brondbjerg posted this on Twitter: You can follow along and see how this idea is evolving into the beautiful, elegant line drawings that are the hallmark of his style. SpheresMike’s post gave me an idea, a way to introduce a concept I like to use in some of my work: signed distance functions. Sdfs are more commonly associated with raytracing and shaders, and by far the best source to learn about them in that context is Inigo Quilez, of Shadertoy fame: https://iquilezles.org/. But there is a delightfully wrong way to use a sdf. Their primary use in raytracing and shaders is to define meshless geometry. For example, a way to draw smooth spheres without generating a ton of triangles. The use I’m showing here is the exact opposite: to generate geometry, point clouds in fact. Geometry that then needs to be processed in a conventional way before it can be rendered on the screen. Let’s imagine we want to tackle something similar to the tweet: particles colliding with a sphere. One way to do this is by calculating the distance of the particle to the center of the sphere, let’s keep it at the origin. For a particle \vec{p} at position (x,y,z) , this distance is given by: {d( \vec{p} )=\sqrt{x.x+y.y+z.z}}. If that distance d is larger than the radius r of the sphere, the particle is outside the sphere, if it’s smaller then it is inside, if it’s exactly the same then it is on the surface.\begin{cases} d( \vec{p} )<r, & \text{if } \vec{p}\text{ is inside} \\ d( \vec{p} )=r , & \text{if } \vec{p}\text{ is on sphere}\\ d( \vec{p} )>r, & \text{if } \vec{p}\text{ is outside} \end{cases}Using this, we can check the particles every step. As long as their distance to the sphere is larger than the radius, they’re fine. If a particle takes a step and its distance becomes smaller or equal, it has hit the sphere. A grid of particles flying into a sphere. Some have collided with the sphere, the rest i...